Method and apparatus for efficient three-dimensional contouring of medical images

ABSTRACT

A technique is disclosed for generating a new contour and/or a 3D surface such as a variational implicit surface from contour data. In one embodiment, B-spline interpolation is used to efficiently generate a new contour (preferably a transverse contour), from a plurality of input contours (preferably, sagittal and/or coronal contours). In another embodiment, a point reduction operation is performed on data sets corresponding to any combination of transverse, sagittal, or coronal contour data prior to processing those data sets to generate a 3D surface such as a variational implicit surface. A new contour can also be generated by the intersection of this surface with an appropriately placed and oriented plane. In this manner, the computation of the variational implicit surface becomes sufficiently efficient to make its use for new contour generation practical.

FIELD OF THE INVENTION

The present invention pertains generally to the field of processing medical images, particularly generating contours for three-dimensional (3D) medical imagery.

BACKGROUND AND SUMMARY OF THE INVENTION

Contouring is an important part of radiation therapy planning (RTP), wherein treatment plans are custom-designed for each patient's anatomy. Contours are often obtained in response to user input, wherein a user traces the object boundary on the image using a computer workstation's mouse and screen cursor. However, it should also be noted that contours can also be obtained via automated processes such as auto-thresholding programs and/or auto-segmentation programs.

FIG. 1 depicts an exemplary GUI 100 through which a user can view and manipulate medical images. The GUI 100 includes frame 102 corresponding to the transverse (T) viewing plane, frame 104 corresponding to the coronal (C) viewing plane, and frame 106 corresponding to the sagittal (S) viewing plane. Within frame 102, an image slice of a patient that resides in a T plane can be viewed. Within frame 104, an image slice of a patient that resides in a C plane can be viewed. Within frame 106, an image slice of a patient that resides in an S plane can be viewed. Using well-known techniques, users can navigate from slice-to-slice and viewing plane-to-viewing plane within GUI 100 for a given set of image slices. It can also be noted that the upper right hand frame of GUI 100 depicts a 3D graphics rendering of the contoured objects.

FIG. 2( a) illustrates an exemplary patient coordinate system with respect to a radiotherapy treatment machine that is consistent with the patient coordinate system defined by the IEC 61217 Standard for Radiotherapy Equipment. As can be seen, the patient coordinate system is a right-hand coordinate system such that if a supine patient is lying on a treatment couch with his/her head toward the gantry, the positive x-axis points in the direction of the patient's left side, the positive y-axis points in the direction of the patient's head, and the positive z-axis points straight up from the patient's belly. The origin of this coordinate system can be offset to the origin of the image data under study.

FIG. 2( b) defines the T/S/C viewing planes with respect to the patient coordinate system of FIG. 2( a). As is understood, a plane in the T viewing plane (the xz-viewing plane) will have a constant value for y, a plane in the S viewing plane (the yz-viewing plane) will have a constant value for x, and a plane in the C viewing plane (the xy-viewing plane) will have a constant value for z.

Returning to the example of FIG. 1, the image data within GUI 100 depicts a patient's prostate 110, bladder 112, and rectum 114. As indicated above, an important part of RTP is the accurate contouring of regions of interest such as these.

Current RTP software typically limits contour drawing by the user through GUI 100 to T views (views which are perpendicular to the patient's long axis) as the T images usually have the highest spatial resolution, the T images are the standard representation of anatomy in the medical literature, and the T contours are presently the only format defined in the DICOM standard. The two other canonical views—the S and C views—can then be reconstructed from the columns and rows, respectively, of the T images.

When generating 3D surfaces from image slices, conventional software programs known to the inventor herein allow the user to define multiple T contours for a region of interest within an image for a plurality of different T image slices. Thereafter, the software program is used to linearly interpolate through the different T contours to generate a 3D surface for the region of interest. However, the inventor herein notes that it is often the case that a plane other than a T plane (e.g., planes within the S and/or C viewing planes) will often more clearly depict the region of interest than does the T plane. Therefore, the inventor herein believes there is a need in the art for a robust 3D contouring algorithm that allows the user to define input contours in any viewing plane (including S and C viewing planes) to generate a 3D surface for a region of interest and/or generate a new contour for the region of interest.

Further still, the inventor herein believes that conventional 3D surface generation techniques, particularly techniques for generating variational implicit surfaces, require unacceptably long computational times. As such, the inventor herein believes that a need exists in the art for a more efficient method to operate on contours in three dimensions.

Toward these ends, according to one aspect of an embodiment of the invention, disclosed herein is a contouring technique that increases the efficiency of 3D contouring operations by reducing the number of data points needed to represent a contour prior to feeding those data points to a 3D contouring algorithm, wherein the 3D contouring algorithm operates to generate a 3D surface such as a variational implicit surface or process the reduced data points to generate a new contour in a new plane via an interpolation technique such as B-spline interpolation. The data points that are retained for further processing are preferably a plurality of shape-salient points for the contour. In accordance with one embodiment, computed curvature values for the data points are used as the criteria by which to judge which points are shape-salient. In accordance with another embodiment, computed scalar second derivative values are used as the criteria by which to judge which points are shape-salient. In accordance with yet another embodiment, the DeBoor equal energy theorem is used as the criteria by which to judge which points are shape-salient.

According to another aspect of an embodiment of the invention, disclosed herein is a contouring technique that operates on a plurality of data points, wherein the data points define a plurality of contours corresponding to a region of interest within a patient, each contour being defined by a plurality of the data points and having a corresponding plane, wherein the plurality of data points are reduced as described above and processed to find the reduced data points that intersect a new plane, and wherein B-spline interpolation is used to interpolate through the points of intersection to generate a new contour in the new plane. This embodiment can operate on a plurality of contours drawn by a user in the S and/or C viewing planes to generate a T contour in a desired T plane. The point reduction operation performed prior to the B-spline interpolation improves the efficiency of the B-spline interpolation operation.

While various advantages and features of several embodiments of the invention have been discussed above, a greater understanding of the invention including a fuller description of its other advantages and features may be attained by referring to the drawings and the detailed description of the preferred embodiment which follow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an exemplary graphical user interface (GUI) for a contouring program, wherein the GUI displays 3D patient Computed Tomography (CT) data in separate planar views;

FIG. 2( a) depicts an exemplary patient coordinate system with respect to a radiotherapy treatment machine;

FIG. 2( b) depicts the T, S, and C view planes for the patient coordinate system of FIG. 2( a);

FIG. 3 depicts an exemplary contour specified by cubic B-splines;

FIG. 4 depicts an exemplary contour and its approximation using cubic B-spline interpolation using eight points on the original contour;

FIGS. 5( a) and (b) depict exemplary computing environments on which embodiments of the present invention can be realized;

FIG. 6 depicts an exemplary process flow for generating a new contour from a plurality of input points that represent contours;

FIG. 7 depicts a plot of curvature versus arc length for sampled points from the exemplary contour of FIG. 4 for two finite difference interval settings;

FIG. 8 depicts the exemplary contour of FIG. 4 reconstructed using B-spline interpolation for several different finite difference interval settings with respect to the curvature computation;

FIG. 9 depicts plots of scalar second derivative versus arc length for two settings of the finite difference interval for the exemplary contour of FIG. 4;

FIG. 10 depicts the exemplary contour of FIG. 4 reconstructed using B-spline interpolation for several different finite difference interval settings with respect to the scalar second derivative computation;

FIG. 11 depict a plot of the DeBoor energy versus arc length and a plot of the cumulative distribution of energy for the exemplary contour of FIG. 4;

FIG. 12 depicts three DeBoor energy reconstructions of the exemplary contour of FIG. 4;

FIGS. 13( a)-(c) depict a graphical reconstruction of T contours using B-spline interpolation in accordance with an embodiment of the invention;

FIG. 14 depicts an exemplary process flow for generating a variational implicit surface from a plurality of input points that represent contours;

FIG. 15 depicts, for a pair of actual manually drawn contours, the non-uniform data sampling, the results of the second derivative shape analysis, and the construction of a complete set of constraints for input to an implicit function computation;

FIG. 16 depicts a variational implicit surface produced from the contours of FIG. 15;

FIGS. 17-19( b) demonstrate the application of an exemplary variational implicit surface method in the contouring of the prostate, bladder, and rectum shown in FIG. 1;

FIG. 20 depicts an exemplary process flow for operating on input point sets wherein one or more of the input sets contains a small number of contour data points; and

FIG. 21 depicts an exemplary process flow wherein both B-spline reconstruction of T contours and variational implicit surface generation from reduced data sets is employed.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

I. Contours:

The embodiments of the present invention address contours. Contours are planar, closed curves C(x,y,z) which can be realized as sets of non-uniformly sampled points along the user-input stroke, {c₁, . . . ,c_(M)} (or sets of points generated by an auto-thresholding and/or auto-segmentation program), wherein the individual points are represented by c_(i)=C(x_(i),y_(i),z_(i)), and wherein M is the number of points in the contour. Points c_(i) in the T planes (xz-planes) have y constant, S contours (yz-planes) have x constant, and C contours (xy-planes) have z constant.

Contours can also be parameterized by a curve length u where the curve C of length L is represented as C(x,y,z)=C(x(u),y(u),z(u))=C(u) where 0≦u≦L and C(0)=C(L).

II. B-Spline Representation Of Contours

When contours exist as discrete points as noted above, it can be useful to represent these points as samples on a continuous curve along which one can interpolate the contour shape at any arbitrary point. B-splines, which can specify arbitrary curves with great exactness, can provide such a representation for contours. (See Piegl, L. A., and Tiller, W., The Nurbs Book, Springer, N.Y., 1996, the entire disclosure of which is incorporated herein by reference). The B-spline description of a curve depends on (1) a set of predefined basis functions, (2) a set of geometric control points, and (3) a sequence of real numbers (knots) that specify how the basis functions and control points are composed to describe the curve shape. Given this information, the shape of C(u) can be computed at any u. Alternatively, given points u′ sampled along C(u), one can deduce a set of B-spline control points and corresponding knots that reconstruct the curve to arbitrary accuracy. Thus, B-splines can be used to interpolate curves or surfaces through geometric points or to approximate regression curves through a set of data points.

B-splines form piecewise polynomial curves along u, delimited by the knots u_(i),i=0, . . . ,m into intervals in which subsets of the basis functions and the control points define C(u). The m+1 knots U={u₀, . . . ,u_(m)} are a non-decreasing sequence of real numbers such that u_(i)≦u_(i+1), for all i.

The p-th degree B-spline basis function, N_(i,p)(u), defined for the i-th knot interval, defines the form of the interpolation. The zero-th order function, N_(i,0)(u), is a step function and higher orders are linear combinations of the lower order functions. The construction of basis functions by recursion is described in the above-referenced work by Piegl and Tiller. A preferred embodiment of the present invention described herein employs cubic (p=3) B-splines.

Basis function N_(i,p)(u) is nonzero on the half-open interval [u_(i),u_(i+p+1)), and for any interval [u_(i),u_(i+1)) at most (p+1) of the basis functions, N_(i−p,p)(u), . . . ,N_(ix,p)(u), are nonzero. A p-th degree, open B-spline curve C(u) with end points u=a,b is defined by

$\begin{matrix} {{{C(u)} = {\sum\limits_{i = 0}^{n}{{N_{i,p}(u)}P_{i}}}},{a \leq u \leq b}} & (1) \end{matrix}$ where the P_(i) are the (n+1) control points, the N_(i,p)(u) are the basis functions, and the knot vector U is defined

$\begin{matrix} {U = \left\{ {\underset{\underset{p + 1}{︸}}{a,\ldots\mspace{11mu},a},u_{p + 1},\ldots\mspace{11mu},u_{m - p - 1},\underset{\underset{p + 1}{︸}}{b,\ldots\mspace{11mu},b}} \right\}} & (2) \end{matrix}$ where a≦u_(p+1)≦u_(p+2), . . . ,≦u_(m−p−1)≦b . This defines an unclosed curve with multiple knots at the end values a=u₀, . . . ,u_(p);b=u_(m−p), . . . ,u_(m). For a spline of degree p with m+1 knots, n+1 control points will be required to specify the shape; for all spline geometries p,n,m are related as m=n+p+1.  (3) Closed curves with coincident start and end points and with C² continuity (continuous curve with continuous first and second derivatives) throughout are defined with uniform knot vectors of the form U={u₀,u₁, . . . ,u_(m)} with n+1(=m−p) control points defined such that the first p control points P₀,P₁, . . . ,P_(p−1) are replicated as the last p control points P_(n−p−1), . . . ,P_(n) which for the cubic (p=3) case means that P₀=P_(n−2),P₁=P_(n−1),P₂=P_(n). This means that there are actually n+1−p unique control points, and that the knots that are actually visualizable on a closed curve are the set u_(p),u_(p+1), . . . , u_(m−p−1).

FIG. 3 shows an exemplary continuous closed curve 300 specified by cubic (p=3) B-splines. Curve 300 is defined by five unique control points P₀,P₁, . . . ,P₄ as shown at the vertices of polygon 302. To generate a closed curve with C² continuity everywhere, p=3 of the control points are replicated, making n=7, and since m=n+p+1, then m+1=12 uniformly spaced knots are required, given by the vector U=(0,1,2,3,4,5,6,7,8,9,10,11) For fixed p,n,U, the curve shape 300 can be changed by moving one or more of the control points P. The locations of the knots are shown as dots on curve 300, wherein the knots u₃-u₇ uniquely span the curve, wherein knots u₀-u₂ coincide with knots u₅-u₇, and wherein knots u₈-u₁₁ coincide with knots u₃-u₆. Thus, as with the control points that must be duplicated for cyclic B-spline curves, so too must some of the knots be duplicated. III. B-Spline Interpolation Of Points In Contours

A useful application of B-splines is to interpolate a smooth curve through a series of isolated points that represent samples of a curve. Global interpolation can be used to determine a set of control points given all the data in the input curve. (See Chapter 9 of the above-referenced work by Piegl and Tiller). Suppose one starts with a set of points {Q_(k)},k=0, . . . ,n on the actual curve, and the goal is to interpolate through these points with a p-degree B-spline curve. Assigning a parameter value ū_(k) to each Q_(k) and selecting an appropriate knot vector U={u₀, . . . ,u_(n)}, one can then set up the (n+1)×(n+1) system of linear equations

$\begin{matrix} {Q_{k} = {{C\left( {\overset{\_}{u}}_{k} \right)} = {\sum\limits_{i = 0}^{n}{{N_{i,p}\left( {\overset{\_}{u}}_{k} \right)}P_{i}}}}} & (4) \end{matrix}$ where the n+1 control points P_(i) are the unknowns. The system can be re-written as Q=AP  (5) where the Q,P are column vectors of the Q_(k) and P_(i), respectively, and where A is the matrix of basis functions. This (n+1)×(n+1) linear system can be solved for the unknown control points P_(i) P=A ⁻¹ Q  (6) by factoring A by LU decomposition instead of inverting matrix A. (See Press, et al., Numerical Recipes in C, 2^(nd) Edition, Cambridge University Press, 1992; Golub, G. H. and Van Loan, C. F., Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996, the entire disclosures of both of which are incorporated herein by reference). A higher quality reconstruction—end points joined with C² continuity—can be obtained by restricting curves to cubic (p=3) type and by specifying endpoint first derivatives. Defining the endpoint tangent vectors D₀ at Q₀ and D_(n) at Q_(n), one constructs a linear system like equation (5) but with two more variables to encode the tangent information resulting in a (n+3)×(n+3) system. The tangents are added to the system with the equations

$\begin{matrix} {{P_{0} = {{Q_{0} - P_{0} + P_{1}} = {{{\frac{u_{4}}{3}D_{0}} - P_{n + 1} + P_{n + 2}} = {\frac{1 - u_{n + 2}}{3}D_{n}}}}}{P_{n + 2} = Q_{n}}} & (7) \end{matrix}$ that can be used to construct a tridiagonal system

$\begin{matrix} {\begin{bmatrix} {Q_{1} - {a_{1}P_{1}}} \\ Q_{2} \\ \vdots \\ Q_{n - 2} \\ {Q_{n - 1} - {c_{n - 1}P_{n + 1}}} \end{bmatrix} = {\begin{bmatrix} b_{1} & c_{1} & 0 & \cdots & 0 & 0 & 0 \\ a_{2} & b_{2} & c_{2} & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{n - 2} & b_{n - 2} & c_{n - 2} \\ 0 & 0 & 0 & \cdots & 0 & a_{n - 1} & b_{n - 1} \end{bmatrix}\begin{bmatrix} P_{2} \\ P_{3} \\ \vdots \\ P_{n - 1} \\ P_{n} \end{bmatrix}}} & (8) \end{matrix}$ that can be solved by Gaussian elimination. (See Chapter 9.2.3 of the above-referenced work by Piegl and Tiller).

To demonstrate the interpolation of points representing a putative curve, the inventor has sampled points from closed curves with random, but known, shapes, and reconstructed the random curves measuring the accuracy as the mean squared error of the reconstructed curve versus the original. FIG. 4 shows an example of a contour 400 from which eight Q_(k) points were selected, and from which a set of control points 412 was computed using equation (8). Contour 400 represents a randomly-generated shape, wherein this shape is approximated by contour 406 based on cubic B-spline interpolation through eight points Q_(k) on the original contour 400. Shown at right in FIG. 4 is a superposition of the approximated contour 406 over the original contour 400. Also shown at right in FIG. 4 is the geometric polygon representation 410 of the spline reconstruction, wherein the newly-determined control points are the vertices 412 of polygon 410, and wherein the knots 408 on the new curve 406 are connected by the line segments to the control points 412. In this example, there are 11 (which equals (n+1)) actual control points 412, which include eight unique control points and the p replicates. Because m=n+p+1=10+3+1=14, the knot vector requires m+1=15 elements for which one can define a uniform (equal knot intervals) knot vector, U=(0,1,2,3,4,5,6,7,8,9,10,11,12,13,14), which is the usual knot configuration for closed curves.

IV. Embodiments Of The Invention

FIGS. 5( a) and 5(b) depict exemplary computing environments in which embodiments of the present invention can be realized. Preferably, a processor 502 is configured to execute a software program to carry out the three-dimensional contouring operations described herein. Such a software program can be stored as a set of instructions on any computer-readable medium for execution by the processor 502. The processor 502 receives as inputs a plurality of data points 500, wherein these input data points 500 are representative of a plurality of contours. These input points can be defined manually by a computer user (e.g., by dragging a mouse cursor over a desired shape to define points for an input contour) or automatically by an auto-thresholding and/or auto-segmentation process, as would be understood by those having ordinary skill in the art.

In the embodiment of FIG. 5( a), the input points 500 are representative of a plurality of contours that reside in at least one viewing plane. Preferably, these contours are a plurality of S contours, a plurality of C contours, or some combination of at least one S contour and at least one C contour. Furthermore, as described hereinafter, the software program executed by processor 502 is preferably configured to generate one or more output contours 504 from the input points 500, wherein the output contour(s) 504 reside in a viewing plane that is non-parallel to the at least one viewing plane for the contours of the input points 500. Preferably, the output contour(s) 504 is/are T contour(s).

FIG. 6 depicts an exemplary process flow for the FIG. 5( a) embodiment. At step 602, the software program receives the plurality of input data points 500. These input points 500 can be grouped as a plurality of different initial sets of data points, wherein each initial point set is representative of a different contour, the different contours existing in at least one viewing plane. As explained above, these contours preferably comprise (1) a plurality of S contours, (2) a plurality of C contours, or (3) at least one S contour and at least one C contour.

One observation that can be made from FIG. 4 is that the reconstructed contour 406 does not match the original contour 400 everywhere. The fit could be improved by sampling more points on the original contour 400, and, in the limit of all available points on the original contour 400, the cubic B-spline reconstruction would be exact. However, the inventor herein notes that some points corresponding to the original contour 400 ought to be more important to contour shape than others—e.g., points at u where the curvature is large should convey more shape information than any other points. Thus, in the interest of increasing the efficiency with which B-spline interpolation can be performed and with which interpolated contours can be represented through data, the inventor herein notes that B-spline interpolation need not be performed on all of the input points 500 for a given contour. Instead, the input points 500 in a given set of input points can be processed to generate a reduced set of input points on which the B-spline interpolation will be performed (step 604). Preferably, step 604 operates to reduce the number of input points 500 for a given contour by generating a reduced set of input points, wherein the reduced set comprises a plurality of shape-salient points. As used herein, points which are representative of a contour are considered “shape-salient” by applying a filtering operation based on a shape-indicative metric to those points, examples of which are provided below.

The inventor herein discloses three techniques that can be used to reduce the data points 500 to a plurality of shape-salient points.

According to a first technique of point reduction for step 604, the shape-salient points for each initial input point set are determined as a function of computed curvature values for a contour defined by the points 500 within that initial input point set. The curvature is representative of the speed at which curve C(u) changes direction with respect to increasing u, wherein u represents the distance along curve C(u) beginning from an arbitrary starting point or origin. The curvature of a plane curve is defined as:

$\begin{matrix} {{\kappa = \frac{{x^{\prime}y^{''}} - {y^{\prime}x^{''}}}{\left\lbrack {x^{\prime\; 2} + y^{\prime\; 2}} \right\rbrack^{3/2}}},} & (9) \end{matrix}$ where x′=dx/du, x″=d²x/du², etc. are derivatives computed by finite differences on uniform u− intervals along C(u), and where the x,y values correspond to points which are representative of the input contour. (See DoCarmo, M., Differential Geometry of Curves and Surfaces, Prentice Hall, N.Y., 1976; Thomas, J. W., Numerical Partial Differential Equations—Finite Difference Methods, Springer, N.Y., 1995, the entire disclosures of which are incorporated herein by reference).

Preferably, step 604 takes points u* at peak values of κ(u) u*=arg max_(u)κ(u)  (10) These points u*, which contribute most importantly to the shape of a curve, are saved for reconstruction of the contour through B-spline interpolation. It should be noted that because of the cyclic nature of the data in u (since 0≦u≦L and C(0)=C(L)), when computing the argmax function over intervals u, one can let the intervals span the origin 0 and then reset the computation for intervals placed at L+a to a or −a to L−a. To accomplish the use of uniform intervals u along C(u), one can (1) reconstruct each input contour via B-spline interpolation through all of its raw input points, (2) step along the reconstructed contour in equal size steps that are smaller than the normal spacing among the raw input points to generate the points which are fed to the curvature computation of formula (9), and (3) apply the curvature computations of formulas (9) and (10) to thereby generate a set of reduced points from the original set of raw input points.

FIG. 7 shows plots of curvature versus arc length u for two settings of the finite difference interval for the random contour 400 of FIG. 4. Plot 700 shows the curvature versus arc length for finite differences over 1.25% of the total curve length, and plot 702 shows the curvature versus arc length for a finite differences of over 5.0% of the total curve length. By varying this interval size, one may detect more or fewer peaks along arc length u; the longer the interval, the more apparent smoothing of the shape, thereby resulting in only the most prominent peaks being detected. For example, in plot 702, only the most prominent directional changes are apparent, leading to a selection of twelve unique Q_(k) points for the B-spline interpolation analysis.

It should also be noted that rather than using only maxima, step 604 can also be configured to retain only those points for which the computed curvature value exceeds a threshold value. As such, it can be seen that a variety of conditions can be used for determining how the curvature values will be used to define the shape-salient points.

FIG. 8 shows the random shape 400 of FIG. 4 (shown with dotted lines in FIG. 8) reconstructed using B-spline interpolation for several interval settings. The number of points Q_(k) for each reconstruction is set by the value of the finite distance interval and also depends on the shape complexity of the curve. For reconstructed contour 802, the number of points Q_(k) was 12. For reconstructed contour 804, the number of points Q_(k) was 16. For reconstructed contour 806, the number of points Q_(k) was 29. As can be seen from FIG. 8, by using more points Q_(k), it is possible to achieve arbitrarily high accuracy. However, it should also be noted that given the original number of samples for this example (a total of 1,099 samples), the use of only 29 points represents a significant compression in the number of points Q_(k) used for the B-spline interpolation while still providing good accuracy in the reconstruction. Optionally, user input can be used to define a setting for the finite difference interval used in the curvature computations. Thus, the finite difference interval size in the curvature calculation can serve as an adjustable tuning parameter which can be controlled to define a quality and efficiency for the contour reconstruction.

According to a second technique of point reduction for step 604, the shape-salient points for each initial input point set are determined as a function of computed scalar second derivative a values (i.e., the scalar acceleration) for the motion of a point along C(u), which is defined as

$\begin{matrix} {{a(u)} = \left\lbrack {\left( \frac{\mathbb{d}^{2}{x(u)}}{\mathbb{d}u^{2}} \right)^{2} + \left( \frac{\mathbb{d}^{2}{y(u)}}{\mathbb{d}u^{2}} \right)^{2}} \right\rbrack^{1/2}} & (11) \end{matrix}$

Preferably, step 604 takes points u* at peak values of a(u), u*=arg max_(u) a(u)   (12) Once again, the derivatives can be computed by finite differences on uniform u− intervals along C(u). Also, as noted above, because of the cyclic nature of the data in u (since 0≦u≦L and C(0)=C(L)), when computing the argmax function over intervals u, one can let the intervals span the origin 0 and then reset the computation for intervals placed at L+a to a or −a to L−a. As with the curvature calculations described above, to accomplish the use of uniform intervals u along C(u), one can (1) reconstruct each input contour via B-spline interpolation through all of its raw input points, (2) step along the reconstructed contour in equal size steps that are smaller than the normal spacing among the raw input points to generate the points which are fed to the scalar second derivative computation of formula (11), and (3) apply the scalar second derivative computation of formulas (11) and (12) to thereby generate a set of reduced points from the original set of raw input points.

FIG. 9 shows plots of scalar second derivative versus arc length u for two settings of the finite difference interval for the random contour 400 of FIG. 4. Plot 900 shows the scalar second derivative versus arc length for finite differences over 5.0% of the total curve length, and plot 902 shows the scalar second derivative versus arc length for a finite differences of over 20.0% of the total curve length. As with the curvature method exemplified by FIG. 7, by varying the finite difference interval size, one may detect more or fewer peaks along arc length u; the longer the interval, the more apparent smoothing of the shape, thereby resulting in only the most prominent peaks being detected. Therefore, plot 902 would be expected to produce fewer Q_(k) points than plot 900.

It should also be noted that rather than using only maxima, step 604 can also be configured to retain only those points for which the computed scalar second derivative exceeds a threshold value. As such, it can be seen that a variety of conditions can be used for determining how the scalar second derivative values will be used to define the shape-salient points.

FIG. 10 shows the random shape 400 of FIG. 4 (shown with dotted lines in FIG. 10) reconstructed using B-spline interpolation for several interval settings. As with the reconstructions of FIG. 8, the number of points Q_(k) for each reconstruction is set by the value of the finite distance interval and also depends on the shape complexity of the curve. For reconstructed contour 1002, the number of points Q_(k) was 10. For reconstructed contour 1004, the number of points Q_(k) was 19. For reconstructed contour 1006, the number of points Q_(k) was 44. As can be seen from FIG. 10, by using more points Q_(k), it is possible to achieve arbitrarily high accuracy. Also, as with the reconstructions of FIG. 8, it should be noted that given the original number of samples for this example (a total of 1,099 samples), the use of only 44 points represents a significant compression in the number of points Q_(k) used for the B-spline interpolation while still providing good accuracy in the reconstruction. Optionally, user input can be used to define a setting for the finite difference interval used in the scalar second derivative computations. Thus, the finite difference interval size in the scalar second derivative calculation can serve as an adjustable tuning parameter which can be controlled to define a quality and efficiency for the contour reconstruction.

According to a third technique of point reduction for step 604, the shape-salient points are determined as a function of the DeBoor equal energy theorem. (See DeBoor, C., A Practical Guide to Splines, Springer, N.Y., 2001, the entire disclosure of which is incorporated herein by reference). With the DeBoor equal energy theorem, the total curvature of the entire curve is divided into s equal parts, and the sampled points are placed along the curve, at s non-uniform intervals, but in such a way as to divide the total curvature into equal parts.

The DeBoor theorem then measures the curvature as the k-th root of absolute value of the k-th derivative of the curve,

$\begin{matrix} {{{D^{k}{C(u)}}}^{1/k} = {{{\frac{\mathbb{d}^{k}}{\mathbb{d}u^{k}}{C(u)}}}^{1/k}.}} & (13) \end{matrix}$ where D^(k)C(u) denotes the derivative operator. The above-referenced work by DeBoor proves two instances of a theorem (Theorem II(20), Theorem XII(34)) that optimally places breakpoints (sample points) to interpolate a curve with minimum error. For a closed curve C(u) of length L such that 0≦u<L, one can define a set of arc length values v_(j), j=1, . . . ,s such that points on C at those values evenly divide the total curvature. The total curvature K is

$\begin{matrix} {K = {\int_{0}^{L}{{{D^{k}{C(u)}}}^{1/k}{\mathbb{d}u}}}} & (14) \end{matrix}$ so that dividing it into s equal parts where the energy of any part is 1/s of the energy of the curve, or

$\begin{matrix} {{\int_{\upsilon_{j}}^{\upsilon_{j + 1}}{{{D^{k}{C(u)}}}^{1/k}{\mathbb{d}u}}} = {\frac{1}{s}{\int_{0}^{L}{{{D^{k}{C(u)}}}^{1/k}{{\mathbb{d}u}.}}}}} & (15) \end{matrix}$ This measure is similar to the ∫(D^(k)C(u))²du “bending energy” curvature measure (see Wahba, G., Spline Models for Observational Data, SIAM (Society for Industrial and Applied Mathematics), Philadelphia, Pa., 1990, the entire disclosure of which is incorporated herein by reference) minimized by spline functions, so it is deemed appropriate to call the DeBoor technique described herein as an “equal energy”theorem or method.

FIG. 11 shows the plot 1102 of the DeBoor energy versus arc length, and a plot 1104 of the cumulative distribution of that energy for the random shape curve 400 of FIG. 4. In this plot, the finite differences were taken across 10% of the curve length.

FIG. 12 shows three DeBoor energy reconstructions of the random shape curve 400 of FIG. 4 (shown as dotted lines in FIG. 12). The number of points Q_(k) for each reconstruction is set by the chosen value for s. For larger values of s, larger values of Q_(k) will result. For reconstructed contour 1202, the number of points Q_(k) was 7. For reconstructed contour 1204, the number of points Q_(k) was 10. For reconstructed contour 1206, the number of points Q_(k) was 20. As can be seen from FIG. 12, the DeBoor equal energy function produces increasingly accurate reconstructions of the random figures as the number of points s is increased. Unlike the curvature and scalar second derivative methods described above, the DeBoor equal energy method fits a given number of points to the curve, instead of the number of points being set by the detection of underlying peaks in the curve. It should be noted, however, for shapes with a high shape complexity or a large value of the integral (14), it may be difficult to anticipate the appropriate number of Q_(k). Optionally, user input can be used to define a setting for s used in the DeBoor computations. Thus, the interval size s in the DeBoor equal energy method can serve as an adjustable tuning parameter which can be controlled to define a quality and efficiency for the contour reconstruction.

Returning to FIG. 6, after a reduced set of points has been generated at step 604 for each initial set of input points 500, step 606 operates to re-order the points within each reduced set with respect to a frame of reference and a direction of rotation. Preferably, the frame of reference is set so that each reduced set's origin (i.e., first) point is at the cranial-most position on the contour. Also, the direction of rotation is preferably set to be clockwise, as determined by the method of Turning Tangents. (See the above-referenced work by DoCarmo). In this manner, for a given reduced point set, the origin point is set as the cranial-most point, and subsequent points are ordered based on a clockwise rotation starting from the origin point. However, it should be understood that other frames of reference and/or directions of rotation could be used.

Next, step 608 operates to find the points of intersection within each reduced set of re-ordered points on a desired new plane (e.g., a plane that is non-parallel to the at least one viewing plane for the contours defined by the reduced sets of input points). Preferably, step 608 operates to find the points within each reduced re-ordered point set that intersect a desired T plane.

After the points of intersection in the new plane (e.g., a T plane) are found, step 610 operates to generate a new contour in this new plane by ordering the points of intersection and interpolating through the points of intersection using B-spline interpolation as described above in Section III.

Thereafter, at step 612, a comparison can be made between the new contour generated at step 610 and a corresponding patient image in the same plane. Such a comparison can be made visually by a user. If the generated contour is deemed a “match” to the image (i.e., a close correspondence between the generated contour and the corresponding anatomy in the displayed image), then the generated contour can be archived for later use (step 614). If the generated contour is not deemed a match to the image, then process flow of FIG. 6 can begin anew with new input points, wherein these new input points perhaps define more contours than were previously used.

FIGS. 13( a)-(c) depict how a set of B-spline T contours can be generated from input points corresponding to three input S contours. FIG. 13( a) depicts a GUI 1300 through which a user can define contours within various CT image slices of a patient. Different frames of the GUI 1300 correspond to different viewing planes into the patient's CT data. Frame 1302 depicts a slice of image data in the T viewing plane. Frame 1304 depicts a slice of image data in the C viewing plane, and frame 1306 depicts a slice of image data in the S viewing plane. The user can navigate from slice-to-slice within the GUI 1300 using conventional software tools. Because the process flow of FIG. 6 allows the user to draw original contours for an anatomical region of interest in any viewing plane, the user can select the viewing plane(s) in which to draw the contours based on which viewing plane(s) most clearly depict the anatomical region of interest. In this example, the user has drawn three contours 1308, 1310, and 1312 in the S viewing plane. The corresponding footprints for these three S contours are shown in frames 1302 and 1304 for the T and C viewing planes respectively. Also, the upper right hand frame of GUI 1300 depicts perspective views of these three S contours. The process flow of FIG. 6 can be invoked to generate a plurality of T contours 1320 from the sampled points for the three S contours 1308, 1310, 1312. FIG. 13( b) depicts several of these generated T contours 1320. Each generated T contour 1320 corresponds to a T contour generated from B-spline interpolation as described in connection with FIG. 6 for a different T plane.

Furthermore, as can be seen in FIG. 13( b), the generated T contours wrap the three original S contours, and a 3D surface 1330 can be rendered from these S and T contours using a series of B-spline interpolations as described above. FIG. 13( c) depicts a rotated view of the contours and surface rendering from FIG. 13( b).

In the embodiment of FIG. 5( b), the input points 500 are representative of a plurality of any combination of T, S, and/or C contours. Furthermore, as described hereinafter, the software program executed by processor 502 in the FIG. 5( b) embodiment is configured to generate a 3D surface 506 from the input points 500. From this 3D surface 506, contours of any obliquity, including T contours, can be readily generated.

FIG. 14 depicts an exemplary process flow for the FIG. 5( b) embodiment. At step 1402, the software program receives the plurality of input points 500 as described above in connection with step 602 of FIG. 6. As with step 602, these input points 500 can be grouped as a plurality of different initial sets of data points 500, wherein each initial point set is representative of a different contour. It should be noted that for a preferred embodiment of the process flow of FIG. 6, the FIG. 6 process flow generates T contours from input contours in the S and/or C viewing planes. However, with a preferred embodiment of the process flow of FIG. 14, the input contours can be in any viewing plane, including T contours.

Next, at step 1404, each initial point set is processed to generate a reduced set of input points, as described above in connection with step 604 of FIG. 6. By compressing the number of points used to represent the different input contours, the computation of the variational implicit surface becomes practical. Without such compression of the contour representations, the computational time for generating the variational implicit surface from the contour representations requires an undue amount of time on conventional computing resources.

Thereafter, at step 1406, a variational implicit surface is generated from the reduced sets of data points. The variational implicit surface is a solution to the scattered data interpolation problem in which the goal is to determine a smooth function that passes through discrete data points. (See Turk and O'Brien, Shape Transformation Using Variational Implicit Functions, Proceedings of SIGGRAPH 99, Annual Conference Series, pp. 335-342, Los Angeles, Calif., August 1999, the entire disclosure of which is incorporated herein by reference). For a set of constraint points {c₁, . . . ,c_(k)} with a scalar height {h₁, . . . ,h_(k)} at each position, one can determine a function ƒ(x),x=(x,y,z)^(T) that passes through each c_(i) such that ƒ(c_(i))=h_(i). A variational solution that minimizes the so-called “bending energy”(see the above-referenced work by Turk and O'Brien) is the sum

$\begin{matrix} {{f(x)} = {{\sum\limits_{j = 1}^{n}{d_{j}{\phi\left( {x - c_{j}} \right)}}} + {P(x)}}} & (16) \end{matrix}$ over radial basis functions φ_(j) (described below) weighted by scalar coefficients d_(j), and where c_(j) are the constraint point locations and P(x) is a degree one polynomial P(x)=p ₀ +p ₁ x+p ₂ y  (17) that accounts for constant and linear parts of the function ƒ(x). The radial basis functions for the 3D constraints appropriate for this problem are φ(x)=|x ³|.  (18) Solving for the constraints h_(i) in terms of the known positions

$\begin{matrix} {h_{i} = {{\sum\limits_{j = 1}^{n}{d_{j}{\phi\left( {c_{i} - c_{j}} \right)}}} + {P\left( c_{i} \right)}}} & (19) \end{matrix}$ gives a linear system that for 3D constraints c_(i)=(c_(i) ^(x),c_(i) ^(y),c_(i) ^(z)) is

$\begin{matrix} {{\begin{bmatrix} \phi_{11} & \phi_{12} & \cdots & \phi_{1k} & 1 & c_{1}^{x} & c_{1}^{y} & c_{1}^{z} \\ \phi_{21} & \phi_{22} & \cdots & \phi_{2k} & 1 & c_{2}^{x} & c_{2}^{y} & c_{2}^{z} \\ \vdots & \vdots & \; & \vdots & 1 & \vdots & \vdots & \vdots \\ \phi_{k\; 1} & \phi_{k\; 2} & \cdots & \phi_{kk} & 1 & c_{k}^{x} & c_{k}^{y} & c_{k}^{z} \\ 1 & 1 & \cdots & 1 & 0 & 0 & 0 & 0 \\ c_{1}^{x} & c_{2}^{x} & \cdots & c_{k}^{x} & 0 & 0 & 0 & 0 \\ c_{1}^{y} & c_{2}^{y} & \cdots & c_{k}^{y} & 0 & 0 & 0 & 0 \\ c_{1}^{z} & c_{2}^{z} & \cdots & c_{k}^{z} & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} d_{1} \\ d_{2} \\ \vdots \\ d_{k} \\ p_{0} \\ p_{1} \\ p_{2} \\ p_{3} \end{bmatrix}} = \begin{bmatrix} h_{1} \\ h_{2} \\ \vdots \\ h_{k} \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}} & (20) \end{matrix}$ This system is symmetric and positive semi-definite, so there will always be a unique solution for the d_(j) and the p_(j). The solution can be obtained using LU decomposition. (See the above-referenced works by Press et al. and Golub and Van Loan). In a preferred embodiment, the implementation of LU decomposition can be the LAPACK implementation that is known in the art. (See Anderson et al., LAPACK User's Guide, Third Edition, SIAM—Society for Industrial and Applied Mathematics, Philadelphia, 1999, the entire disclosure of which is incorporated herein by reference).

A further feature of the variational implicit surface computation as described in the above-referenced work by Turk and O'Brien is the use of additional constraint points, located off the boundary along normals connecting with the on-boundary constraints, to more accurately and reliably interpolate the surface through the on-boundary constraints. In a preferred embodiment, the on-boundary constraints' h_(j) values can be set to 0.0 and the off-boundary values can be set to 1.0. However, as should be understood, other values can be used in the practice of this embodiment of the invention.

FIG. 15 demonstrates, for a pair of actual manually drawn contours 1502 and 1504, the non-uniform data sampling, the results of the DeBoor energy analysis using 20 points with both contours, and the construction of a complete set of constraints for input to the implicit function computation. FIG. 15 depicts the original contour points 1506 for an S contour 1502 and a C contour 1504. Also depicted in FIG. 15 are the shape-salient points 1508 (shown as the darker points along the contours) computed from the original points 1506 using the above-described DeBoor equal energy technique. Furthermore, FIG. 15 depicts the normals 1510 (shown as boxes) to the on-contour constraint points 1508. The implicit function constraint points would thus include both the on-contour shape-salient points 1508 and their corresponding normals 1510.

Performance of this solution depends partly on the form of the radial basis function φ(x)one uses, and on the size of the system parameterk (number of all constraint points). The performance of the LU solution of equation (20) can be done using different choices of φ. (See Dinh, et al., Reconstructing surfaces by volumetric regularization using radial basis functions, IEEE Transactions on Pattern Analysis and Machine Intelligence, 24, pp. 1358-1371, 2002, the entire disclosure of which is incorporated herein by reference). The function |x³| is monotonic increasing, meaning that the matrix in (20) has large off-diagonal values for all constraint point pairs c_(i),c_(j),i≠j. To make the linear system perform more robustly, the above-referenced work by Dinh describes a modification of the system to make it more diagonally dominant by adding to the diagonal elements a set of scalar values λ_(i)

$\begin{matrix} {{\begin{bmatrix} {\phi_{11} + \lambda_{1}} & \phi_{12} & \cdots & \phi_{1k} & 1 & c_{1}^{x} & c_{1}^{y} & c_{1}^{z} \\ \phi_{21} & {\phi_{22} + \lambda_{2}} & \cdots & \phi_{2k} & 1 & c_{2}^{x} & c_{2}^{y} & c_{2}^{z} \\ \vdots & \vdots & \; & \vdots & 1 & \vdots & \vdots & \vdots \\ \phi_{k\; 1} & \phi_{k\; 2} & \cdots & {\phi_{kk} + \lambda_{k}} & 1 & c_{k}^{x} & c_{k}^{y} & c_{k}^{z} \\ 1 & 1 & \cdots & 1 & 0 & 0 & 0 & 0 \\ c_{1}^{x} & c_{2}^{x} & \cdots & c_{k}^{x} & 0 & 0 & 0 & 0 \\ c_{1}^{y} & c_{2}^{y} & \cdots & c_{k}^{y} & 0 & 0 & 0 & 0 \\ c_{1}^{z} & c_{2}^{z} & \cdots & c_{k}^{z} & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} d_{1} \\ d_{2} \\ \vdots \\ d_{k} \\ p_{0} \\ p_{1} \\ p_{2} \\ p_{3} \end{bmatrix}} = {\begin{bmatrix} h_{1} \\ h_{2} \\ \vdots \\ h_{k} \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}.}} & (20) \end{matrix}$ A preferred embodiment uses values λ_(Boundary)=0.001 and λ_(Offboundary)=1.0. However, it should be understood that other values could be used.

After solving for the d_(j) and the p_(j) in Equation (20), the implicit function in (16) can be evaluated to determine that set of points {x_(i)} for which ƒ(x_(i))=0. (The zero-th level of ƒ(x) is that on which the boundary points lie). The method of Bloomenthal (see Bloomenthal, J., An Implicit Surface Polygonizer, Graphics Gems IV, P. Heckbert, Ed., Academic Press, New York, 1994, the entire disclosure of which is incorporated herein by reference) can be used to track around the function and determine the locations of mesh nodes from which a 3D surface may be constructed. A closed surface constructed in this way can be termed a variational implicit surface. (See the above-referenced work by Turk and O'Brien).

At step 1408, that mesh can then be clipped by planes parallel to the xz-plane at the appropriate y-value(s) to produce the desired T contour(s) for display to the user. The mesh representation and clipping functionality can be performed using the VTK software system available from Kitware, Inc. of Clifton Park, N.Y. (See Schroeder et al., The Visualization Toolkit, 4^(th) Ed., Kitware, 2006, the entire disclosure of which is incorporated herein by reference).

Thereafter, as with steps 612 and 614 of FIG. 6, the generated contour can be compared to the image (step 1410) and archived if it sufficiently matches the anatomy of interest shown in the image (step 1412). If not a sufficient match, the process of FIG. 14 can begin anew.

As indicated, the main performance limitation for computing a variational implicit surface is the total number of constraints, and for k greater than a few thousand, the variational implicit surface computation takes too much time to be useful for real-time applications. However, the inventor herein believes that by reducing the number of constraint points used for computing the variational implicit surface via any of the compression operations described in connection with steps 1404 and 604 for contour representations, the computation of variational implicit surfaces will become practical for 3D medical contouring. Furthermore, a fortunate property of the variational implicit surface is its ability to forgive small mismatches in orthogonal contours that are required to intersect (because they are curves on the same surface) but do not because the user was unable to draw them carefully enough. For example, when the sampling interval in the Bloomenthal algorithm is set to the inter-T plane distance, the resulting surfaces are sampled at too coarse a level to reveal the small wrinkles in the actual surface, and the resulting T contours are not affected by the missed T/S/C intersections.

FIG. 16 illustrates the variational implicit surface 1602 produced from the contours 1502 and 1504 in FIG. 15. FIG. 16 also depicts cutaways 1604 and 1606 that show the profiles of the contours 1502 and 1504, respectively, that were used to compute surface 1602.

FIGS. 17-19( b) demonstrate the application of the variational implicit surface method in the contouring of three organs shown in FIG. 1—the prostate 110, bladder 112, and rectum 114. In FIG. 17, variational implicit contours of the prostate 110 are shown in wireframe along with a conventional T-only contoured rendering. The top part of FIG. 17 depicts a left sagittal view of the variational implicit contours. The bottom part of FIG. 17 depicts a frontal (anterior) coronal view of the variational implicit contours. The wireframe for FIG. 17 was created using a single C contour and three S contours as inputs.

In FIG. 18, variational implicit contours of the bladder 112 are shown in wireframe along with a conventional T-only contoured rendering. The top portion of FIG. 18 depicts a right sagittal view of the variational implicit contours, and the bottom portion of FIG. 18 depicts a frontal (anterior) coronal view of the variational implicit contours. The wireframe for FIG. 18 was generated using a single C contour and three S contours as inputs.

In FIGS. 19( a) and (b), variational implicit contours of the rectum 114 are shown in wireframe along with a conventional T-only contoured rendering. FIG. 19( a) depicts a left sagittal view of the variational implicit contours, and FIG. 19( b) depicts a frontal (anterior) coronal view of the variational implicit contours. The wireframe for FIGS. 19( a) and (b) was generated using a three C contours and five S contours as inputs.

As shown by FIGS. 17-19( b), the wireframes show good agreement with the conventionally drawn structures depicted at the right in these figures.

It should also be noted that it may sometimes be the case wherein an initial set of input points corresponding to a contour contains only a small number of points, long gaps in the sequence of input points, and/or two or more points having the same coordinates (x,y). For example, both a small number of points and long gaps between points would likely result when a user defines a contour by only picking points at the vertices of a polygon that approximates the contour. Duplicate points can result when the user picks points along a contour because a graphics subsystem will sometimes interpret a single mouse button push as multiple events. In such instances, the process flow of FIG. 20 can be employed. At step 2002, an input point set for a contour is received. At step 2004, a check is made as to whether there are points with duplicate (x,y) coordinates. If yes at step 2004, the program proceeds to step 2006, where all duplicate points are removed from the contour's point set. If not, the input point set is retained and the process proceeds to step 2008. At step 2008, the program determines whether there are gaps in the input point set greater than a threshold fraction of the total contour length. If yes at step 2008, the program proceeds to step 2010 which augments the input point set by filling the gaps with points by linear interpolation between the input points at the end points of the gap. Optionally, the identities of the original points from the input point set and the identifies of the points added at step 2010 can be preserved so that the process of finding the shape-salient points within the point set will only allow the original points to be classified as shape-salient. If no at step 2008, then the program proceeds to step 2012. At step 2012, a contour is interpolated from the points in the input point set using B-spline interpolation as previously described in Section III. Thereafter, at step 2014, a plurality of points can be sampled from the interpolated contour, and these sampled points can be used to replace and/or augment the points in the input point set used to represent that contour. Step 2016 then operates to check whether the user has defined any additional contours. If not, the process flow can proceed to further complete the contouring operations (such as by proceeding to step 604 of FIG. 6 or step 1404 of FIG. 14). If the user has provided further input, the process can return to step 2002.

It should also be noted that the B-spline interpolation and variational implicit surface generation can be combined in a single process flow as different modes of operation, as shown in FIG. 21. At step 2102, various input point sets for different contours are received. Then, at step 2104, the process flow decides which reconstruction mode should be used—e.g., a B-spline interpolation mode or a variational implicit surface mode. The decision at step 2104 can be made in any of a number of ways. For example, the B-spline interpolation mode can be used where only a small number (e.g., less than or equal to three) of S and/or C input contours have been defined, and the variational implicit surface mode can be used for other cases. Further still, a user can select which mode to be used via some form of user mode input.

If the B-spline interpolation mode is used, then steps 2106 and 2108 can be performed, wherein these steps correspond to steps 606 and 608 from FIG. 6, albeit without the preceding point reduction operation of step 604. However, it should also be noted that steps 2106 and 2108 could be replaced by steps 604, 606, and 608 if desired by a practitioner of this embodiment of the invention. Step 610 preferably operates as described above in connection with FIG. 6.

If the variational implicit surface mode is used, then steps 1404, 1406, and 1408 can be followed as described in connection with FIG. 14.

While the present invention has been described above in relation to its preferred embodiments, various modifications may be made thereto that still fall within the invention's scope. Such modifications to the invention will be recognizable upon review of the teachings herein. Accordingly, the full scope of the present invention is to be defined solely by the appended claims and their legal equivalents. 

1. An apparatus for generating a contour corresponding to a region of interest within an image, the apparatus comprising: a processor configured to (1) compute a plurality of scalar second derivative values for a first plurality of data points, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, (2) define a second plurality of data points as a function of the computed scalar second derivative values, the second plurality being less than the first plurality, and (3) generate a new contour in a new plane based on the second plurality of data points, the new contour corresponding to the region of interest.
 2. The apparatus of claim 1 wherein the processor is further configured to (1) generate a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (2) clip the 3D surface at the new plane to generate the new contour.
 3. An apparatus for generating a contour corresponding to a region of interest within an image, the apparatus comprising: a processor configured to (1) generate a second plurality of data points from a first plurality of data points, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, (2) find the points within the second plurality of points that intersect a new plane, and (3) generate a new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection using B-spline interpolation, the new contour corresponding to the region of interest.
 4. The apparatus of claim 3 wherein the processor is further configured to (1) compute a plurality of curvature values for the first plurality of data points, and (2) define the second plurality of data points as a function of the computed curvature values.
 5. The apparatus of claim 3 wherein the processor is further configured to (1) compute a plurality of scalar second derivative values for the first plurality of data points, and (2) define the second plurality of data points as a function of the computed scalar second derivative values.
 6. The apparatus of claim 3 wherein the processor is further configured to define the second plurality of data points from the first plurality of data points based on a DeBoor equal energy theorem function.
 7. The apparatus of claim 3 wherein the processor is further configured to (1) re-order the second plurality of data points with respect to a frame of reference and a direction of rotation, and (2) find the points that intersect the new plane using the re-ordered second plurality of data points.
 8. An apparatus for generating a contour corresponding to a region of interest within an image, the apparatus comprising: a processor configured to (1) define a second plurality of data points from a first plurality of data points based on a DeBoor equal energy theorem function, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, and (2) generate a new contour in a new plane based on the second plurality of data points, the new contour corresponding to the region of interest.
 9. The apparatus of claim 8 wherein the processor is further configured to (1) generate a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (2) clip the 3D surface at the new plane to generate the new contour.
 10. The apparatus of claim 9 wherein the new plane comprises a plane of any user-selected obliquity.
 11. The apparatus of claim 9 wherein the new plane comprises a transverse plane.
 12. The apparatus of claim 8 wherein the corresponding plane comprises a sagittal plane, and wherein the new plane comprises a transverse plane.
 13. The apparatus of claim 8 wherein the corresponding plane comprises a coronal plane, and wherein the new plane comprises a transverse plane.
 14. The apparatus of claim 8 wherein the corresponding plane comprises a plurality of corresponding planes, the plurality of corresponding planes comprising a sagittal plane and a coronal plane, and wherein the new plane comprises a transverse plane.
 15. The apparatus of claim 8 wherein the processor is further configured to (1) select between a plurality of contour reconstruction modes, the plurality of contour reconstruction modes comprising a first contour reconstruction mode and a second contour reconstruction mode, the first contour reconstruction mode being configured to (a) find the points within the second plurality of points that intersect the new plane, and (b) generate the new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection using B-spline interpolation, the second contour reconstruction mode being configured to (a) generate a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (b) clip the 3D surface at the new plane to generate the new contour, and (2) perform the selected contour reconstruction mode to thereby generate the new contour.
 16. The apparatus of claim 15 wherein the processor is further configured to automatically select between the first and second contour reconstruction modes based on how many contours are represented by the first plurality of data points.
 17. The apparatus of claim 15 wherein the processor is further configured to select between the first and second contour reconstruction modes based on input from a user.
 18. The apparatus of claim 8 wherein the processor is further configured to (1) receive a plurality of initial point sets, each initial point set corresponding to a contour and comprising a plurality of input points, and (2) process the input points of the received initial point sets to generate the first plurality of data points.
 19. The apparatus of claim 18 wherein the processor is further configured to process the input points of the received initial point sets to generate the first plurality of data points by removing duplicate points from the initial point sets.
 20. The apparatus of claim 18 wherein the processor is further configured to process the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) detecting whether a gap is present between the input points of that initial point set that are greater than a threshold fraction of a total length for the contour corresponding to that initial point set, and (2) in response to detecting that a gap is present, filling in each detected gap by linearly interpolating between the input points of that initial point set that correspond to end points of that detected gap.
 21. The apparatus of claim 18 wherein the processor is further configured to process the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) interpolating another contour from that initial point set by performing B-spline interpolation on the input points of that initial point set, (2) sampling a plurality of points on the another contour, and (3) using the sampled points as part of the first plurality of data points rather than the input points of that initial point set.
 22. The apparatus of claim 18 wherein the processor is further configured to process the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) detecting whether any duplicate points are present in that initial point set, (2) in response to detecting that a duplicate point is present in that initial point set, removing each duplicate point from that initial point set, (3) detecting whether a gap is present between the input points of that initial point set that are greater than a threshold fraction of a total length for the contour corresponding to that initial point set, (4) in response to detecting that a gap is present, filling in each detected gap by linearly interpolating between the input points of that initial point set that correspond to end points of that detected gap, (5) interpolating another contour from that initial point set by performing B-spline interpolation on the input points and any filled points of that initial point set, (6) sampling a plurality of points on the another contour, and (7) using the sampled points as part of the first plurality of data points rather than the input points of that initial point set.
 23. A computer-readable storage medium for generating a contour corresponding to a region of interest within an image, the computer-readable storage medium comprising: a plurality of computer-executable instructions for (1) computing a plurality of scalar second derivative values for the first plurality of data points, (2) defining a second plurality of data points as a function of the computed scalar second derivative values, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, and (3) generating a new contour in a new plane based on the second plurality of data points, the new contour corresponding to the region of interest, and wherein the instructions are resident on the computer-readable storage medium.
 24. The computer-readable storage medium of claim 23 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) generating a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (2) clipping the 3D surface at the new plane to generate the new contour.
 25. A computer-implemented method for generating a contour corresponding to a region of interest within an image, the method comprising: generating a second plurality of data points from a first plurality of data points based on a DeBoor equal energy theorem function, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane; and generating a new contour in a new plane based on the second plurality of data points, the new contour corresponding to the region of interest; and wherein the method steps are performed by a processor.
 26. The method of claim 25 wherein the new contour generating step comprises the following steps performed by the processor: generating a three-dimensional (3D) surface that is representative of the region of interest based on the second plurality of data points; and clipping the 3D surface at the new plane to generate the new contour.
 27. The method of claim 26 wherein the 3D surface generating step comprises the processor generating a variational implicit surface that is representative of the region of interest based on the second plurality of data points.
 28. The method of claim 27 wherein the new plane comprises a plane of any user-selected obliquity.
 29. The method of claim 28 wherein the new plane comprises a transverse plane.
 30. The method of claim 25 wherein the new contour generating step comprises the following steps performed by the processor: finding the points within the second plurality of points that intersect the new plane; and generating the new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection using B-spline interpolation.
 31. The method of claim 30 wherein the corresponding plane comprises a sagittal plane, and wherein the new plane comprises a transverse plane.
 32. The method of claim 30 wherein the corresponding plane comprises a coronal plane, and wherein the new plane comprises a transverse plane.
 33. The method of claim 30 wherein the corresponding plane comprises a plurality of corresponding planes, the plurality of corresponding planes comprising a sagittal plane and a coronal plane, and wherein the new plane comprises a transverse plane.
 34. A computer-implemented method for generating a contour corresponding to a region of interest within an image, the method comprising: generating a second plurality of data points from a first plurality of data points, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, wherein the first plurality of points comprises a plurality of initial point sets, each initial point set comprising a plurality of points from the first plurality of points and being representative of one of the contours corresponding to the region of interest, and wherein the step of generating the second plurality of points comprises performing a point reduction operation separately on each of the initial point sets to generate a plurality of corresponding reduced point sets; and generating a new contour in a new plane based on the plurality of corresponding reduced point sets, the new contour corresponding to the region of interest; and wherein the method steps are performed by a processor.
 35. The method of claim 34 wherein the step of generating the second plurality of points comprises: the processor reducing the first plurality of points to a plurality of shape-salient points by performing the point reduction operation separately on each of the initial point sets to generate the plurality of corresponding reduced point sets.
 36. The method of claim 34 further comprising the following method steps performed by the processor: computing a plurality of curvature values for the initial point sets; and generating the reduced point sets as a function of the computed curvature values.
 37. The method of claim 34 further comprising the following method steps performed by the processor: computing a plurality of scalar second derivative values for the initial point sets; and generating the reduced point sets as a function of the computed scalar second derivative values.
 38. The method of claim 34 further comprising the following method step performed by the processor: generating the reduced point sets based on a DeBoor equal energy theorem function.
 39. The method of claim 34 further comprising the following method step performed by the processor: controlling how many points are present in the reduced point sets via a user-adjustable tuning parameter.
 40. A computer-implemented method for generating a contour corresponding to a region of interest within an image, the method comprising: generating a second plurality of data points from a first plurality of data points, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane; finding the points within the second plurality of points that intersect a new plane; and generating a new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection, the new contour corresponding to the region of interest; and wherein the method steps are performed by a processor.
 41. The method of claim 40 wherein the interpolating step comprises the processor interpolating through the points of intersection using B-spline interpolation to generate the new contour.
 42. A computer-readable storage medium for generating a contour corresponding to a region of interest within an image, the computer-readable storage medium comprising: a plurality of computer-executable instructions for (1) defining a second plurality of data points from a first plurality of data points based on a DeBoor equal energy theorem function, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, and (2) generating a new contour in a new plane based on the second plurality of data points, the new contour corresponding to the region of interest, and wherein the instructions are resident on the computer-readable storage medium.
 43. The computer-readable storage medium of claim 42 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) generating a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (2) clipping the 3D surface at the new plane to generate the new contour.
 44. The computer-readable storage medium of claim 43 wherein the new plane comprises a plane of any user-selected obliquity.
 45. The computer-readable storage medium of claim 43 wherein the new plane comprises a transverse plane.
 46. The computer-readable storage medium of claim 42 wherein the corresponding plane comprises a sagittal plane, and wherein the new plane comprises a transverse plane.
 47. The computer-readable storage medium of claim 42 wherein the corresponding plane comprises a coronal plane, and wherein the new plane comprises a transverse plane.
 48. The computer-readable storage medium of claim 42 wherein the corresponding plane comprises a plurality of corresponding planes, the plurality of corresponding planes comprising a sagittal plane and a coronal plane, and wherein the new plane comprises a transverse plane.
 49. The computer-readable storage medium of claim 42 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) selecting between a plurality of contour reconstruction modes, the plurality of contour reconstruction modes comprising a first contour reconstruction mode and a second contour reconstruction mode, the first contour reconstruction mode being configured to (a) find the points within the second plurality of points that intersect the new plane, and (b) generate the new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection using B-spline interpolation, the second contour reconstruction mode being configured to (a) generate a three-dimensional (3D) variational implicit surface that is representative of the region of interest based on the second plurality of data points, and (b) clip the 3D surface at the new plane to generate the new contour, and (2) performing the selected contour reconstruction mode to thereby generate the new contour.
 50. The computer-readable storage medium of claim 49 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for automatically selecting between the first and second contour reconstruction modes based on how many contours are represented by the first plurality of data points.
 51. The computer-readable storage medium of claim 49 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for selecting between the first and second contour reconstruction modes based on input from a user.
 52. The computer-readable storage medium of claim 42 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) receiving a plurality of initial point sets, each initial point set corresponding to a contour and comprising a plurality of input points, and (2) processing the input points of the received initial point sets to generate the first plurality of data points.
 53. The computer-readable storage medium of claim 52 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for processing the input points of the received initial point sets to generate the first plurality of data points by removing duplicate points from the initial point sets.
 54. The computer-readable storage medium of claim 52 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for processing the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) detecting whether a gap is present between the input points of that initial point set that are greater than a threshold fraction of a total length for the contour corresponding to that initial point set, and (2) in response to detecting that a gap is present, filling in each detected gap by linearly interpolating between the input points of that initial point set that correspond to end points of that detected gap.
 55. The computer-readable storage medium of claim 52 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for processing the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) interpolating another contour from that initial point set by performing B-spline interpolation on the input points of that initial point set, (2) sampling a plurality of points on the another contour, and (3) using the sampled points as part of the first plurality of data points rather than the input points of that initial point set.
 56. The computer-readable storage medium of claim 52 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for processing the input points of the received initial point sets to generate the first plurality of data points by, for each initial point set, (1) detecting whether any duplicate points are present in that initial point set, (2) in response to detecting that a duplicate point is present in that initial point set, removing each duplicate point from that initial point set, (3) detecting whether a gap is present between the input points of that initial point set that are greater than a threshold fraction of a total length for the contour corresponding to that initial point set, (4) in response to detecting that a gap is present, filling in each detected gap by linearly interpolating between the input points of that initial point set that correspond to end points of that detected gap, (5) interpolating another contour from that initial point set by performing B-spline interpolation on the input points and any filled points of that initial point set, (6) sampling a plurality of points on the another contour, and (7) using the sampled points as part of the first plurality of data points rather than the input points of that initial point set.
 57. A computer-readable storage medium for generating a contour corresponding to a region of interest within an image, the computer-readable storage medium comprising: a plurality of computer-executable instructions for (1) generating a second plurality of data points from a first plurality of data points, the second plurality being less than the first plurality, the first plurality of data points being representative of a plurality of contours corresponding to the region of interest, each contour having a corresponding plane, (2) finding the points within the second plurality of points that intersect a new plane, and (3) generating a new contour in the new plane based on the second plurality of data points by interpolating through the points of intersection using B-spline interpolation, the new contour corresponding to the region of interest, and wherein the instructions are resident on the computer-readable storage medium.
 58. The computer-readable storage medium of claim 57 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) computing a plurality of curvature values for the first plurality of data points, and (2) defining the second plurality of data points as a function of the computed curvature values.
 59. The computer-readable storage medium of claim 57 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) computing a plurality of scalar second derivative values for the first plurality of data points, and (2) defining the second plurality of data points as a function of the computed scalar second derivative values.
 60. The computer-readable storage medium of claim 57 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for defining the second plurality of data points from the first plurality of data points based on a DeBoor equal energy theorem function.
 61. The computer-readable storage medium of claim 57 wherein the instructions further comprise computer-executable instructions resident on the computer-readable storage medium for (1) re-ordering the second plurality of data points with respect to a frame of reference and a direction of rotation, and (2) finding the points that intersect the new plane using the re-ordered second plurality of data points. 